Approximation, significant figures and scientific notation

IB Mathematics Analysis and Approaches SL· Unit 1: Number & Algebra· 15 min read

1. Significant Figures: Counting and Rounding★☆☆☆☆⏱ 5 min

📘 Definition

Significant figures (sig figs)

sf

Digits in a number that indicate the precision of a measurement. Leading zeros are never significant; trailing zeros are only significant if there is a decimal point.

Example:

0.0023 has 2 sf, 1200 has 2 sf, 120.0 has 4 sf

Counting significant figures follows clear rules that you must memorize for the exam, as questions will explicitly ask you to round to a given number of sf:

Non-zero digits are always significant
Any zero between two non-zero digits is significant
Leading zeros (before the first non-zero) are never significant
Trailing zeros after a decimal point are always significant
Trailing zeros in a whole number with no decimal point are not counted as significant

📐 Worked Example

Count the number of significant figures in each value: (a) 0.04010, (b) 15000, (c) 203.0

1
Apply sig fig rules to (a): leading zeros before 4 are not significant. All digits after the first non-zero are significant, so:
2
$0.04010 \rightarrow 4 \text{ sf}$
3
For (b): trailing zeros in the whole number 15000 have no decimal point, so only 1 and 5 are significant:
4
$15000 \rightarrow 2 \text{ sf}$
5
For (c): the zero between 2 and 3 is significant, and the trailing zero after the decimal is significant, so all 4 digits count:
6
$203.0 \rightarrow 4 \text{ sf}$

📐 Worked Example

Round 12.478 to 3 significant figures

1
Identify the third significant figure (4), then check the next digit, which is 7
2
Since 7 ≥ 5, round the third digit up from 4 to 5
3
$12.478 \rightarrow 12.5 \text{ (3 sf)}$

2. Scientific Notation★☆☆☆☆⏱ 5 min

📘 Definition

Scientific notation

a \times 10^n

A standardized way to write any number, where $1 \leq a < 10$ and $n$ is an integer. It removes ambiguity about the number of significant figures for large/small values.

Example:

15000 written to 3 sf is $1.50 \times 1 0^{4}$

Scientific notation makes it easy to work with extremely large or small values, and clearly communicates the number of significant figures regardless of the size of the number.

📐 Worked Example

Write 0.0004250 in scientific notation, preserving all significant figures

1
Move the decimal point until exactly one non-zero digit is to the left of the point. For 0.0004250, this gives $a = 4.250$
2
Count how many places the decimal moved: it moved 4 places to the right, so $n = - 4$
3
All digits in $a$ are significant, matching the original value
4
$0.0004250 = 4.250 \times 10^{-4}$

📐 Worked Example

Convert $3.14 \times 1 0^{5}$ to standard decimal form

1
$n = 5$ is positive, so move the decimal 5 places to the right
2
Add zeros to fill empty places
3
$3.14 \times 10^5 = 314000$

3. Approximation in IB Exams★★☆☆☆⏱ 5 min

You will always need to apply approximation skills in exams, whether a question explicitly asks for it or not. Incorrect precision is one of the most common causes of lost marks, even when your method is correct.

📐 Worked Example

Estimate $\frac{39.7 \times 102}{1.97}$ , giving your answer to 1 significant figure

1
Round each value to 1 significant figure for estimation:
2
$39.7 \approx 40, \quad 102 \approx 100, \quad 1.97 \approx 2$
3
Calculate with the rounded values:
4
$\frac{40 \times 100}{2} = 2000$
5
2000 is already to 1 significant figure, so this is the final answer

4. Common Pitfalls

Wrong move:

Counting leading zeros as significant figures

Why:

Leading zeros only mark the position of the decimal point, they do not contribute to precision

Correct move:

Ignore all zeros before the first non-zero digit when counting sig figs

Wrong move:

Forgetting trailing zeros after a decimal point are significant

Why:

A trailing zero after a decimal shows the measurement is precise to that position

Correct move:

Count all trailing zeros after a decimal point, so 2.50 has 3 significant figures

Wrong move:

Writing scientific notation with a coefficient ≥ 10 (e.g. $10 \times 1 0^{3}$ )

Why:

The coefficient must always satisfy $1 \leq a < 10$ by definition

Correct move:

Adjust the coefficient and exponent to put a in the valid range: $10 \times 1 0^{3} = 1 \times 1 0^{4}$

Wrong move:

Rounding intermediate steps in a multi-step calculation

Why:

Early rounding creates accumulated error that leads to an incorrect final answer

Correct move:

Keep full precision during calculation, only round the final answer to the required number of sig figs

Wrong move:

Ignoring the requested number of sig figs in the question

Why:

Examiners deduct marks for incorrect precision even if the value is close

Correct move:

Always check the question for the required precision before writing your final answer

5. Quick Reference Cheatsheet

Rule Type	Description	Example
Non-zero digits	Always significant	123 → 3 sf
Zeros between non-zeros	Always significant	103 → 3 sf
Leading zeros	Never significant	0.012 → 2 sf
Trailing after decimal	Always significant	1.20 → 3 sf
Scientific notation form	$1 \leq a < 10$ , $n$ is integer	$1500 = 1.5 \times 1 0^{3}$
IB default rule	3 sf for all non-exact answers	N/A

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Round value to 3 significant figures
2022 · 2
Convert large value to scientific notation
2023 · 1
Count significant figures in decimal

Going deeper

syllabusOfficial IB AA SL SyllabusFull unit 1 syllabus reference

What's Next

Mastering approximation and significant figures is a foundational skill that applies to every topic in IB AA SL, from quadratic equations to calculus and statistics. Incorrect rounding or precision is one of the most common causes of lost marks in exams, even when your core method for solving a problem is fully correct. Taking the time to lock in these rules now will save you marks across the entire syllabus. Next, you will build on these approximation skills to learn about percentage error when working with measured values, before moving on to other core number and algebra topics.